Auslander [1]} proved that, if \((M,N)\)  is Tor-independent (i.e., \(\operatorname{Tor}_i^R(M,N)=0\)  for all \(i\ge 1\) ) and \(\operatorname{pd}_R(M)<\infty \) , then the depth formula holds for \((M,N)\) .
Auslander's result has been extended by Huneke and Wiegand for complete intersection rings: Tor-independent modules over complete intersection rings satisfy the depth formula; see [2]}.
More generally, one has:
